Module categories over the Drinfeld double of a finite group

TL;DR

This work classifies indecomposable module categories over the Drinfeld center ${\mathcal{D}}(G,\omega)$ of the twisted group category ${\rm Vec}^G_{\omega}$, reducing the problem to a finite group–cohomology data problem. Using dualities between ${\mathcal{C}}$ and ${\mathcal{C}}^{*}$, and the relation between bimodule categories and algebras $A(H,\psi)$, the author proves that indecomposable module categories are parametrized by conjugacy classes of pairs $(H,\psi)$ with $H\subset G\times G$ satisfying $\tilde{\omega}|_H=1$, where $\tilde{\omega}=p_1^{*}\omega-p_2^{*}\omega$. The main theorem specializes existing duality results to the Drinfeld center, and the paper provides a detailed analysis in the concrete case $G=S_3$, including subgroup enumerations, counts of indecomposable module categories for various twists, and a discussion of modular invariants and open problems. The results advance understanding of boundary conditions in holomorphic orbifold models and contribute to the structure theory of modular tensor categories arising from finite group doubles.

Abstract

We classify the module categories over the double (possibly twisted) of a finite group.